Wavelets provide an alternative to classical Fourier methods for one- and multi-dimensional data analysis and synthesis, and have numerous applications both within mathematics and in areas as diverse as physics, seismology, medical imaging, digital image processing, signal processing and computer graphics and video.
A class of wavelet transforms was proposed by Daubechies in 1988. These transforms have signal processing properties that are complementary to Fourier transforms and provide an efficient way to characterize asynchronous signals. These transforms are typically performed on discrete time or sampled data. The simplistic view of the transform is that the signal is applied to a low pass and high pass filter, resulting in a detail signal output from the high pass filter, and an approximation signal output from the low pas filter. This transform can be recursively applied to the approximation signal, resulting in what is called a multi-resolutional analysis. One important characteristic of this transform is that the original signal can be recreated from the collection of detail signals and the lowest level approximation signal. The high and low pass filters have specific characteristics that allow an ideal reconstruction to occur. One of the several characteristics is that the filters, as viewed in the z domain, are Finite Impulse Response (FIR) filters with no transmission poles.
While wavelet transforms can provide an important role in signal processing, one requirement is that the signal be digitally sampled in order to apply the transform. At frequencies where efficient sampling hardware does not exist, there is a need for an approach whereby a wavelet transform can be performed without digitizing the signal. The present invention provides a solution to meet such need.